In earlier work, we described a ``pathological'' example of a signature scheme that is secure in the random-oracle model, but for which no secure implementation exists. For that example, however, it was crucial that the scheme is able to sign "long messages" (i.e., messages whose
length is not a-priori bounded). This left open the possibility that
the Random Oracle Methodology is sound with respect to signature schemes
that sign only "short" messages (i.e., messages of a-priori bounded length, smaller than the length of the keys in use), and are "memoryless" (i.e., the only thing kept between different signature
generations is the initial signing-key). In this work, we extend our negative result to address such signature schemes. A key ingredient in our proof is a new type of interactive proof systems, which may be of independent interest.